596 research outputs found
Random wavelet series based on a tree-indexed Markov chain
We study the global and local regularity properties of random wavelet series
whose coefficients exhibit correlations given by a tree-indexed Markov chain.
We determine the law of the spectrum of singularities of these series, thereby
performing their multifractal analysis. We also show that almost every sample
path displays an oscillating singularity at almost every point and that the
points at which a sample path has at most a given Holder exponent form a set
with large intersection.Comment: 25 page
Describability via ubiquity and eutaxy in Diophantine approximation
We present a comprehensive framework for the study of the size and large
intersection properties of sets of limsup type that arise naturally in
Diophantine approximation and multifractal analysis. This setting encompasses
the classical ubiquity techniques, as well as the mass and the large
intersection transference principles, thereby leading to a thorough description
of the properties in terms of Hausdorff measures and large intersection classes
associated with general gauge functions. The sets issued from eutaxic sequences
of points and optimal regular systems may naturally be described within this
framework. The discussed applications include the classical homogeneous and
inhomogeneous approximation, the approximation by algebraic numbers, the
approximation by fractional parts, the study of uniform and Poisson random
coverings, and the multifractal analysis of L{\'e}vy processes.Comment: 94 pages. Notes based on lectures given during the 2012 Program on
Stochastics, Dimension and Dynamics at Morningside Center of Mathematics, the
2013 Arithmetic Geometry Year at Poncelet Laboratory, and the 2014 Spring
School in Analysis held at Universite Blaise Pasca
Multivariate Davenport series
We consider series of the form , where
and is the sawtooth function. They are the natural multivariate
extension of Davenport series. Their global (Sobolev) and pointwise regularity
are studied and their multifractal properties are derived. Finally, we list
some open problems which concern the study of these series.Comment: 43 page
Large intersection properties in Diophantine approximation and dynamical systems
We investigate the large intersection properties of the set of points that
are approximated at a certain rate by a family of affine subspaces. We then
apply our results to various sets arising in the metric theory of Diophantine
approximation, in the study of the homeomorphisms of the circle and in the
perturbation theory for Hamiltonian systems.Comment: 24 page
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